Dynamic Time Warping Framework
⋆Lu Bai1, Luca Rossi2, Lixin Cui1, and Edwin R. Hancock3 1
Central University of Finance and Economics, Beijing, China 2Aston University, Birmingham, UK
3
University of York, York, UK
Abstract. In this paper, we propose a novel nested alignment graph kernel draw-ing on depth-based complexity traces and the dynamic time warpdraw-ing framework. Specifically, for a pair of graphs, we commence by computing the depth-based complexity traces rooted at the centroid vertices. The resulting kernel for the graphs is defined by measuring the global alignment kernel, which is developed through the dynamic time warping framework, between the complexity traces. We show that the proposed kernel simultaneously considers the local and global graph characteristics in terms of the complexity traces, but also provides richer statistic measures by incorporating the whole spectrum of alignment costs be-tween these traces. Our experiments demonstrate the effectiveness and efficiency of the proposed kernel.
1
Introduction
In pattern recognition, graph kernels are powerful tools for applying standard machine learning techniques to graph datasets [24]. These kernels are typically used in conjuc-tion with kernel methods such as Support Vector Machines (SVM) and kernel Principle Component Analysis (kPCA) for the purposes of classification or clustering [4, 21].
The idea underpinning most existing graph kernels is that of decomposing graph-s into graph-subgraph-structuregraph-s and comparing pairgraph-s of graph-specific igraph-somorphic graph-subgraph-structuregraph-s. Some examples are graph kernels based on counting pairs of isomorphic a) walks [27], b) paths [1], and c) restricted subgraph or subtree substructures [14]. Other examples in-clude the work of Bach [2], who proposed a family of kernels for comparing point clouds. These kernels are based on a local tree-walk kernel between subtrees, which is defined by a factorization on suitably defined graphical models of the subtrees. Wang and Sahbi [28], on the other hand, defined a graph kernel for action recognition. They first describe actions in the videos using directed acyclic graphs (DAGs). The result-ing kernel is defined as an extendresult-ing random walk kernel by countresult-ing the number of isomorphic walks of DAGs. Harchaoui and Bath [18] proposed a segmentation graph kernel for images by counting the inexact isomorphic subtree patterns between image segmentation graphs. Other state-of-the-art graph kernels include the subtree-based hy-pergraph kernel [7], the Lov´asz graph kernel [19], the aligned subgraph kernel [10], the
⋆
subgraph matching kernel [21], the fast depth-based subgraph kernel [6], the optimal assignment kernel [22], and the aligned Jensen-Shannon subgraph kernel [11].
Unfortunately, all the aforementioned graph kernels tend to capture only local char-acteristics of graphs, since they usually use substructures of limited sizes. As a result, these kernels may fail to reflect global graph characteristics. To overcome this short-coming, Johansson et al. [19] developed a family of global graph kernels using geomet-ric embeddings. Specifically, they use the Lov´asz number and its associated orthonor-mal representation to capture global graph characteristics. Bai et al. and Rossi et al. [4, 9, 26, 25] developed a family of graph kernels based on the classical Jensen-Shannon divergence, as well as its quantum analogue. Specifically, they use either the classical or the quantum walk together with quantum information theoretical measures to probe the global structure of the graph.
The aim of this work is to overcome the gap between local kernels (i.e., kernels based on local substructures of limited sizes) and the global kernels (i.e., global ker-nels and quantum or classical Jensen-Shannon kerker-nels), by proposing a novel nested alignment kernel for graphs based on their depth-based complexity traces [5] and the dynamic time warping framework [15]. For a pair of graphs, we commence by com-puting the depth-based complexity traces rooted at the centroid vertices. The resulting kernel is defined by measuring the global alignment kernel [15] between the complex-ity traces. Recall that the depth-based complexcomplex-ity trace of a graph is based on a family of expansion subgraphs that form a nested sequence which gradually expands from the centroid vertex to the global graph structure. As a consequence, this sequence of sub-graphs can reflect both local and global structure information of a graph. Furthermore, we show that the associated global alignment kernel encapsulates the whole spectrum of the alignment cost between the complexity traces. As a result, the proposed kernel can not only simultaneously consider both local and global graph characteristics in terms of the nested depth-based complexity traces, but also provide richer statistic measures by incorporating the whole spectrum of alignment costs between these traces. Experiments demonstrate the effectiveness and efficiency of the proposed kernel.
The remainder of this paper is organized as follows. Section 2 reviews the pre-liminary concepts that will be used in this work. Specifically, we introduce the global alignment kernel through the dynamic time warping framework and the depth-based complexity trace. Section 3 defines the proposed nested alignment kernel. Section 4 provides the experimental evaluation. Section 6 concludes this work.
2
Preliminary Concepts
In this section, we review some preliminary concepts that will be used in this work. We commence by reviewing the dynamic time warping framework. Specifically, we introduce the global alignment kernel based on this framework. Finally, we review the concept of depth-based complexity trace of a graph.
2.1 Global Alignment Kernels from the Dynamic Time Warping Framework
In this subsection, we review the global alignment kernel based on the dynamic time warping framework proposed in [15]. Let Tbe a set of discrete time series that take
values in a space X. For a pair of discrete time seriesP = (p1, . . . , pm) ∈ Tand
Q= (q1, . . . , qn)∈Twith lengthsmandnrespectively, the alignmentπbetweenP
andQis defined as a pair of increasing integral vectors(πp, πq)of lengthl≤m+n−1,
where
1 =πp(1)≤ · · · ≤πp(l) =m
and
1 =πq(1)≤ · · · ≤πq(l) =n
such that(πp, πq)is defined to have unitary increments and no simultaneous repetitions.
For any index1≤i≤l−1, the increment vector ofπ= (πp, πq)satisfies
( πp(i+ 1)−πp(i) πq(i+ 1)−πq(i) ) ∈ {( 0 1 ) , ( 1 0 ) , ( 1 1 )} . (1)
In the dynamic time warping framework [15], the coordinatesπp andπq of the
align-mentπdefine the warping function. LetA(m, n)be the set of all possible alignments betweenPandQ. The dynamic time warping distance betweenPandQis defined as DTW(P,Q) = minπ∈A(m,n)DP,Q(π), (2) where the cost
DP,Q(π) = |π|
∑
i=1
φ(pπp(i), qπq(i)), (3)
is defined by a local divergenceφthat measures the discrepancy between any pair of elementspi ∈ Pandqi ∈ Q. Generally,φcan be defined as the squared Euclidean
distance, i.e.,φ(p, q) =∥p−q∥2.
Based on the dynamic time warping distance defined in Eq.(2), a dynamic time warping kernelkDTW[17] betweenPandQcan be defined as
kDTW(P,Q) =e−DTW(P,Q). (4) Unfortunately, this kernel is not positive definite. This is because the optimal alignment required by the dynamic time warping cannot guarantee transitivity. To overcome the shortcoming, Cuturi [15] considers all possible alignments inA(m, n)and proposes another dynamic time warping inspired kernel, i.e., the global alignment kernel, as
kGA(P,Q) = ∑
π∈A(m,n)
e−DP,Q(π), (5)
wherekGAis positive definite, since it quantifies the quality of both the optimal align-ment and all other alignalign-mentsπ∈ A(m, n). The kernelkGAelaborates on the dynamic time warping distance by considering the same set of elementary operations [16]. How-everkGAnot only generalizes the dynamic time warping kernelkDTW, but also pro-vides richer statistic measures by incorporating the whole spectrum of alignment costs
{DP,Q(π), π∈ A(m, n)}.
Intuitively, the global alignment kernelkGAallows one to define a new graph ker-nel, by measuring the warping alignmentπbetween any types of graph characteristic
sequences (or graph embedding vectors [13])) that have certain element orders with in-creasing structural variables, e.g, the depth-based complexity traces [5] from expansion subgraphs of increasing sizes, or cycle characteristics with increasing lengths identified from the Ihara zeta function [23].
2.2 Centroid Depth-based Complexity Traces
We review the concept of the depth-based complexity trace of a graph rooted at the centroid vertex [5]. LetG(V, E)be an undirected graph with vertex setV and edge setE. Based on Dijkstra’s algorithm, we commence by computing the shortest path matrixSG, where each elementSG(v, u)ofSG represents the length of the shortest
path between verticesv ∈ V andu ∈ V. For each vertex v ∈ V, let S(v)be the average length of the shortest paths fromvto the remaining vertices, i.e.,
S(v) = 1
|V| ∑
u∈V
SG(v, u). (6)
As discussed in [5], the centroid vertexˆvCofG(V, E)can be identified by selecting the
vertex that has the minimum variance of shortest path lengths to the remaining vertices, i.e., the index ofˆvCis
ˆ vC= arg min v ∑ u∈V [SG(v, u)−SV(v)]2. (7) LetNˆvK
C be a vertex subset ofG(V, E)satisfying
NˆvK
C ={u∈V |SG(ˆvC, u)≤K}. (8)
ForG(V, E)and its centroid vertexˆvC, we construct a family ofK-layer expansion
subgraphsGK(VK;EK)as { VK ={u∈NvˆKC}; EK ={(u, v)⊂NvˆKC ×N K ˆ vC |(u, v)∈E}. (9) Note that the number expansion subgraphs is equal to the greatest lengthLof the short-est paths from the centroid vertex to the remaining vertices ofG(V, E). Moreover, the L-layer expansion subgraph is the graphG(V, E)itself. An example of constructing a K-layer subgraph is shown in Fig.1.
Definition (Depth-based complexity traces):For a sample undirected graphG(V, E), let{G1,· · ·,GK,· · ·,GL}be the family ofK-layer expansion subgraphs rooted at the
centroid vertex ofG(V, E). Then the depth-based complexity traceDB(G)ofG(V, E) is computed by measuring the entropies of the subgraphs [5], i.e.,
DB(G) ={HS(G1),· · · , HS(GK),· · · , HS(GL)}, (10)
where· · ·, HS(GK)is the Shannon entropy associated with the steady state random
Fig. 1.The left-most figure shows the determination ofK-layer centroid expansion subgraphs for a graphG(V, E)which hold|Nvˆ1C|= 6and|N
2 ˆ
vC|= 10vertices. While the middle and the
right-most figure show the corresponding1-layer and2-layer subgraphs regarding the centroid vertexˆvC, and are depicted by red-colored edges. In this example, the vertices of differentK -layer subgraphs regarding the centroid vertexˆvCare calculated by Eq.(7), and pairwise vertices possess the same connection information in the original graphG(V, E).
The depth-based complexity trace has a number of interesting properties [5]. First, it encapsulates the entropy-based information content flow through the family ofK-layer expansion subgraphs rooted at the centroid vertex, and thus reflects rich intrinsic depth topology information of a graph. Second, it can be efficiently computed also on large graphs. This is because it is computed on a small set of expansion subgraphs rooted at the centroid vertex, and the computational complexity is polynomial. Furthermore, based on Eq.(9), we can also observe that the family ofK-layer expansion subgraphs rooted at the centroid vertexvˆC of the graphGconstructs a nested sequence. This is
because the family of the expansion subgraphs satisfies ˆ
vC∈ G1· · · ⊆ GK ⊆ · · · ⊆ GL⊆G.
In other words, it represents a sequence of subgraphs that gradually expand from the centroid vertex to the global graph. As a result of it nested nature, the depth-based complexity trace can reflecs both the local and global structure information of a graph. In summary, the depth-based complexity trace provides an elegant way of developing novel fast graph kernels that simultaneously consider local and global graph structures.
3
The Proposed Kernel
In this section, we introduce a novel nested alignment graph kernel through the dynamic time warping framework and the depth-based complexity trace.
3.1 A Nest Aligned Kernel from the Dynamic Time Warping Framework
LetGP(VP, EP)andGQ(VQ, EQ)be a pair of graphs, from a graph setG. We
com-mence by computing the depth-based complexity traces ofGP andGQas
and
DB(GQ) ={HS(GQ;1),· · · , HS(GQ;K),· · ·, HS(GQ;Lmax)},
respectively. HereGP;K andGQ;K are theK-layer expansion subgraphs rooted at the
centroid vertices ofGP andGQ, andLmaxis the greatest length of the shortest paths
rooted at the centroid vertices over all graphs inG. Note that, forGP andGQand the
greatest lengthsM andNof the shortest paths rooted at their centroid vertices, ifK≥ M andK ≥ M their K-layer expansion subgraphs are themselves, i.e., their global structures. Based on the global alignment kernel defined in Section 2.1, we develop a new nested alignment graph kernelkNAbetweenGPandGQas
kNA(GP, GQ) =kGA(DB(GP),DB(GQ))
= ∑
π∈A(Lmax,Lmax)
e−DP,Q(π), (11)
whereπdenotes the warping alignment betweenDB(GP)andDB(GQ),A(Lmax, Lmax)
denotes all possible alignments, andDP,Q(π)is the alignment cost defined in Eq.(3). Note that we cannot prove that the the proposed kernelkNA is positive definite. Al-though our kernel is based on the global alignment kernelkGA, which is a positive definite kernel, the time series compared by kNA are not defined over the same un-derlying space but on two different graphs. Future work will explore the possibility of creating a positive definite kernel by computing the depth-based complexity traces over a common structure obtained by combining the input graphs.
As we have observed, the depth-based complexity trace reflects the nested entropy-based information and thus simultaneously considers the local and global graph struc-tures. Furthermore, the proposed kernelkNA(GP, GQ)is based on all possible warping
alignments between depth-based complexity traces of the input graphs. As a result, kNA(GP, GQ)can simultaneously capture richer local and global graph characteristics
in terms of all possible alignments between the nested depth-based complexity traces.
3.2 Computational Analysis
For a pair of graphs both havingnvertices, computing the nested alignment kernelkGA has time complexity O(n3). This is because computing the depth-based complexity trace of a graph relies on the computation of the shortest path matrix and thus has time complexityO(n3). Furthermore, computing all possible alignments between the depth-based complexity traces has time complexityO((Lmax)2), whereLmaxis the greatest
length of the shortest paths rooted at the centroid vertices of the two graphs and is lower than the vertex numbern. As a result, the proposed kernelkGA has polynomial time complexityO(n3).
4
Experimental Evaluations
4.1 Graph Datasets
We evaluate our kernels on standard graph datasets. These datasets include: MUTAG, PTC, COIL5, Shock and CATH2. Details of these datasets are shown in Table 1.
Table 1.Information on the selected graph based bioninformatics datasets Datasets MUTAG PTC COIL Shock CATH2 Max # vertices 28 109 241 33 568
Min # vertices 10 2 72 4 143
Mean # vertices 17.93 25.60 144.90 109.63 308.03
# graphs 188 344 360 150 190
# classes 2 2 5 5 2
MUTAG: The MUTAG dataset consists of graphs representing 188 chemical com-pounds labeled according to whether or not they affect the frequency of genetic mu-tations in the bacterium Salmonella typhimuriums and aims to predict whether each compound is associated with mutagenicity.
PTC: The PTC (The Predictive Toxicology Challenge) dataset records the carcino-genicity of several hundred chemical compounds for male rats (MR), female rats (FR), male mice (MM) and female mice (FM). These graphs are very small, i.e.,20−30 vertices, and sparsem, i.e.,25−40edges. We select the graphs of male rats (MR) for evaluation. There are344test graphs in the MR class.
COIL5:The COIL5 dataset is abstracted from the COIL image database. The COIL database consists of images of 100 3D objects. In our experiments, we use the images for the first five objects. For each of these objects we employ 72 images captured from different viewpoints. For each image we first extract corner points using the Harris de-tector, and then establish Delaunay graphs based on the corner points as vertices. Each vertex is used as the seed of a Voronoi region, which expands radially with a constant speed. The linear collision fronts of the regions delineate the image plane into polygons, and the Delaunay graph is the region adjacency graph for the Voronoi polygons.
Shock:The Shock dataset consists of graphs from the Shock 2D shape database. Each graph is a skeletal-based representation of the differential structure of the boundary of a 2D shape. There are 150 graphs divided into 10 classes.
CATH2:The CATH2 dataset is harder to classify, since the proteins in the same topol-ogy class are structurally similar. The protein graphs are 10 times larger in size than chemical compounds, with 200 . 300 vertices. There is 190 testing graphs in the dataset.
5
Experiments on Standard Graph Datasets
We evaluate the performance of the nested alignment graph kernel (NAGK) on a number of graph classification tasks. Furthermore, we also compare our kernel with three state-of-the-art kernels, including 1) the Jensen-Shannon graph kernel (JSGK) [4], 2) the random walk graph kernel (RWGK) [20], 3) the unaligned quantum Jensen-Shannon graph kernel (QJSK) [9], and 4) the Lov´asz graph kernel (LGK) [19].
We compute the kernel matrix associated with each kernel on each dataset. We per-form10-fold cross-validation using a C-Support Vector Machine (C-SVM) to compute the classification accuracies, using LIBSVM software library [12]. We use nine sam-ples for training and one for testing. The parameters of the C-SVMs are optimized on each training set using cross-validation. We report the average classification accuracy
and the runtime for each kernel in Table 2 and Table 3. The runtime is measured under Matlab R2015a running on a2.5GHz Intel2-Core processor (i.e., i5-3210m).
Table 2.Classification Accuracy (In%±Standard Error) Runtime in Second.
Datasets MUTAG PTC COIL5 Shock CATH2
NAGK 84.22±.50 58.00±.64 69.75±.65 37.60±.62 74.00±.83
JSGK 83.11±.80 57.29±.41 69.13±.79 21.73±.76 72.26±.76
RWGK 80.77±.75 53.97±.31 14.21±.65 0.33±.37 −
QJSK 82.72±.44 56.70±.49 70.11±.61 40.60±.92 71.11±.88
LGK 80.83±.43 56.29±.47 − 31.80±.89 −
Table 3.Runtime for Various Kernels.
Datasets MUTAG PTC COIL5 Shock CATH2 NAGK 8.6·102 2.3·103 3.3·103 3.8·102 9.4·102
JSGK 1.0·100 1.0·100 1.0·100 1.0·100 1.0·100
RWGK 4.6·101 6.7·101 1.1·103 2.3·101 −
QJSK 2.0·101 1.0·102 1.0·103 1.4·101 4.4·103
LGK 1.0·103 7.4·103 − 1.0·103 −
In terms of classification accuracy, Table 2 indicates that the proposed NAGK ker-nel can significantly outperform the alternative state-of-the-art graph kerker-nels, excluding the QJSK kernel on the COIL5 and Shock datasets. However, the proposed NAGK k-ernel is still competitive to the QJSK kk-ernel on the COIL5 dataset and outperforms the QJSK kernel on the MUTAG, PTC and CATH2 datasets. The reasons for this effective-ness are twofold. First, as we have stated, the depth-based complexity traces used by the proposed NAGK kernel encapsulate nested entropy-based information that extend from the centroid vertex to the global graph structure. As a consequence, the proposed NAGK kernel can simultaneously consider the local and global graph characteristics. By contrast, the the QJSK and JSGK kernels can only reflect global graph characteris-tics, whereas the LGK and RWGK can only reflect local graph characteristics. Second, the proposed NAGK kernel is based on all possible alignments between the complexi-ty traces, and thus reflects rich statistic measures by incorporating the whole spectrum of alignment costs. On the other hand, we observe that the QJSK kernel based on the global von Neumann entropy from the continuous-time quantum walk is the most com-petitive kernel to the proposed NAGK kernel, though the QJSK kernel can only reflect global characteristics. This is because the entropy measure from the quantum walk can reflect richer intrinsic topology information than that from the classical steady state random walk (for the proposed NAGK kernel). This in turn suggest the possibility of further extending the NAGK kernel using quantum walks to extract an analogous of the depth-based complexity trace used in this study.
In terms of runtime, the proposed the NAGK kernel is not the fastest kernel, when compared to the other graph kernels. However, we can observe that the proposed NAGK kernel can always complete the computation of the kernel matrices, unlike some alter-native graph kernels (e.g., the LGK and RWGK kernels), which failed complete the computation in a reasonable time.
6
Conclusion
In this paper, we have proposed a novel nested alignment graph kernel. The kernel is an adaptation of the dynamic time warping framework based kernel (i.e., the global align-ment kernel) to graphs. To this end, we made use of the depth-based complexity traces of graphs, a powerful and fast to compute graph descriptor. Unlike most existing graph kernels that only probe local or global graph characteristics, the proposed kernel simul-taneously considers local and global graph characteristics and thus reflects the presence of richer structural patterns. The experiments have demonstrated the effectiveness and efficiency of the proposed kernel.
Our future work is to extend the proposed kernel to attributed graphs that encap-sulate vertex and edge labels. Moreover, we would also like to further develop nov-el graph kernnov-els through the dynamic time warping framework associated with oth-er types of (hypoth-er)graph charactoth-eristic sequences, e.g., the cycle numboth-ers identified by the Ihara zeta function, the time-varying entropies computed from the continuous-time or discrete-continuous-time quantum walk [9, 8], and the depth-based hypergraph complexity traces [3]. Finally, we are also interested in developing novel graph kernels for time-varying financial market networks [29], using the dynamic time warping framework.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 61503422 and 61602535), the Open Projects Program of National Laboratory of Pattern Recognition, the Y-oung Scholar Development Fund of Central University of Finance and Economics (No. QJJ1540), and the program for innovation research in Central University of Finance and Economics.
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